#498534

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14,17,21)(15,23,25,37)(16,28,30,43)(18,31,29,22)(19,32,33,26)(20,35,36,24)(27,42,41,39)(34,40,38,44);(1,3,4,2)(6,9)(7,11);(13,15,24,40,18,27,30,32,17,25,35,44,29,41,16,26)(14,19,28,39,22,34,36,37,21,33,43,42,31,38,20,23);(5,7,8,11)(6,10,9,12)(15,25)(19,33)(23,37)(26,32)(27,41)(34,38)(39,42)(40,44)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405730);